Graduate Degree

Programs in

 

Applied Mathematics

 

University of Colorado at Boulder

 

 

Academic Year 2005-2006

 

 

SUPPLEMENT TO THE CATALOG

 

 

 

 

 

 

 

Department of Applied Mathematics

University of Colorado

526 UCB

Boulder, Colorado   80309-0526

Phone: (303) 492-1238

email: appm_info@colorado.edu

http://amath.colorado.edu/

Introduction

 

The Department of Applied Mathematics of the University of Colorado at Boulder offers a range of courses and research opportunities in several areas including:

 

Computational Mathematics

Nonlinear Phenomena

Physical Applied Mathematics

Probability and Statistics

 

For more information on these areas and research specialties of the Department and affiliated faculty, please refer to our web site.  The following document provides information on a variety of topics, ranging from admission procedures to degree requirements.  If you have questions that are not answered by this document, please do not hesitate to contact us.

 

 

Graduate Committee

 

The Graduate Committee is responsible for interpretation and administration of the policies described in this supplement.  By its own initiative or by petition from the student or the student's advisor, the Graduate Committee may consider exceptions to these policies when warranted by special circumstances.  The Graduate Committee currently consists of Professors Thomas Manteuffel (Chair), Jem Corcoran, Keith Julien, Manuel Lladser, and Steve McCormick.

 

 

Admissions

 

The admission requirements spelled out below apply equally to the M.S. and Ph.D. programs.  The program in which the applicant applies for is not, in and of itself, a factor in decisions regarding admission or support.  Applicants should be aware, however, that both programs are competitive and meeting the requirements does not guarantee admission.  Successful applicants will, in general, have records considerably stronger in breadth and quality than these minimum standards suggest.

 

Both Domestic and International applications should be sent to:

Graduate Program Assistant

Department of Applied Mathematics

University of Colorado

526 UCB

Boulder, Colorado 80309-0526

 

 

General Admission Requirements

Applicants for graduate study in Applied Mathematics must hold at least a bachelorŐs degree or its equivalent from an institution comparable to the University of Colorado in applied mathematics, engineering, mathematics, or a natural science.  Applicants with strong backgrounds in the physical sciences are encouraged to apply.  Applicants should have a significant background in mathematical coursework and the mathematical maturity to understand basic concepts in pure and applied mathematics.

 

Exceptional students who have some deficiencies in their mathematics background may also be considered for admission; however, such students will likely have to take some upper division undergraduate mathematics courses during their first year of graduate study (see "Provisional Admission" section).

 

Mathematics Courses

The applicantŐs academic background should include several semesters of upper division courses of a mathematical nature beyond calculus, including advanced calculus and linear algebra.  Additional recommended courses include partial differential equations, complex analysis, numerical analysis, and perhaps probability and statistics.  These courses need not be in a mathematics department; however, they should require the mathematical maturity expected of a strong upper-level mathematics undergraduate.

 

It is also recommended that applicants have some computing experience.  For example, a working knowledge of a scientific programming language like FORTRAN, C, C++, or MATLAB is recommended, and experience with UNIX can be very helpful.

 

GRE Scores

All applicants are required to submit scores from the aptitude portion of the Graduate Record Exam.  The subject portion of the GRE's is not required.

 

TOEFL Scores

All foreign applicants are required to submit scores from the Test of English as a Foreign Language (TOEFL).  This requirement will only be waived when the applicant's native language is English, or if the applicant has received a degree from an English speaking college in the United States.  Minimum TOEFL scores are: 173 (computer-based), 500 (paper), or 61 (internet-based – approximate score).  TSE (Test of Spoken English) is not required.

 

Provisional Admission

Applicants having the listed qualifications will, if accepted, be classified as regular degree students.  Applicants with a few deficiencies in the admission requirements listed above will be considered on an individual basis and, if accepted, will be given provisional status.

 

Financial Aid

Financial aid is available to M.S. and Ph.D. students in the form of teaching assistantships, research assistantships, and fellowships.  Request for financial support should be included with the application for admission.  Applicants are encouraged to apply for need based aid through the Office of Financial Aid at (303) 492-5091.

 

Deadlines

Graduate applications for the next fall semester should be received from domestic applicants by February 1 of that year and from foreign applicants by December 1 of the previous year.  Applicants are strongly encouraged to apply for fall admission.  Due to the structure of the graduate courses, admission to the graduate programs in spring semesters is discouraged.

 

ADVISING

Each new student will be assigned a Faculty Advisor (usually the Chair of the Graduate Committee) for consultation in planning a sound program of study.  Advising includes the courses to be taken and the areas in which to take the preliminary exams.  Incoming students will be prevented from registering until they have approval from their Faculty Advisor.  For Ph.D. students, the Chair of the studentŐs Thesis Committee will assume the duties of the Faculty Advisor when it is formed.

 

 

 

 

                                                 M.S. Degree Requirements

 

General MS Degree Requirements

The University of Colorado Catalog gives the general requirements for the degree of Master of Science in all departments of the University of Colorado.  The following is a description of the requirements that specifically pertain to students pursuing a course of study leading to the degree of Master of Science in Applied Mathematics.  It supplements the requirements in the Catalog.  In all cases not specifically mentioned below, the general requirements as stated in the Catalog are understood to apply.

 

The M.S. degree can be useful as a stepping stone for any student who is considering a Ph.D. program here or elsewhere.  By the same token, it can serve to increase a Ph.D. studentŐs options here.  However, the M.S. is unique and an important program in its own right.  One of its principal advantages is in preparation for teaching or industry, which is the genesis of the required numerical analysis and out-of-department sequences discussed below.  It is also a flexible program in that it supports special interest directions, including the teacher training option listed below.

 

Department Course Requirements

The Department requires a candidate to complete an approved program of study consisting of at least 30 semester hours.  At least 18 of these 30 hours must be in Applied Mathematics courses at the 5000 level or above.  (Neither 4350/5350, 4360/5360, nor 4720/5720 generally count towards this requirement.)  All students must complete two year-long sequences in applied mathematics.  See the "Graduate Courses" section of this supplement for a list of acceptable 5000-level sequences.  Other sequences require Faculty Advisor approval.  If APPM 5600/5610 is not taken, then the numerical preliminary exam becomes compulsory.

 

All students are required to take a year-long sequence in an area where mathematics has significant applications or from the Math Department (Faculty Advisor approval required).  In addition, upon approval by petition to the Graduate Committee, up to 6 hours may be taken in courses at the 4000 level in other departments, provided members of the graduate faculty teach those courses.  M.S. students must maintain a grade point average of 3.0 or better.

 

Transfer Credit

MasterŐs degree students may request a maximum of 9 semester hours to be transferred from another institution.  All transfer requests must have approval of both the Graduate Committee and the Graduate School. Work already applied toward a graduate degree received from CU-Boulder or another institution cannot be accepted for transfer toward another graduate degree of the same level at CU-Boulder.  For example, work already applied to meet requirements for a masterŐs degree earned by a student cannot be used toward a second masterŐs degree from CU.  In addition, work completed for a doctoral degree may not be applied toward a subsequent masterŐs degree.

 

Foreign Language Requirement

There is no foreign language requirement for the M.S. program.

 

Thesis vs. Non-thesis Option

The MasterŐs degree requirements may be fulfilled by following the requirements for either the thesis (Plan I) or the non-thesis (Plan II) option (see the Catalog for details).  Students who elect Plan II must have approval of the Chair of the Graduate Committee.

 

A student electing to do a thesis must take 4-6 hours of thesis credit, which count towards the required 30 hours, and must take an oral comprehensive exam on his/her thesis work.  This will be administered by a committee consisting of the Faculty Advisor and two other faculty members, at least one of whom must be a regular or affiliate member of the Department.  The student is expected to submit a hard copy (bound, on thesis bond) and an electronic copy (in PostScript or PDF format) of the thesis to the Department before graduation.  The electronic version will be placed on the Department web site for public access. The department copy of the dissertation should have title, author, and graduation year embossed on the spine.  To help cover this additional expense, the department will reimburse the student up to $30 toward the binding expense.  The Denver Bookbinding Company (http://www.denverbook.com/) provides this service at a reasonable cost.

 

A student electing the non-thesis option must pass (Pass or Ph.D. Research Pass) any one of the four Ph.D. preliminary exams.  (See the "Ph.D. Degree Requirements" section for more detail).

 

A student who fails either the written preliminary exam or the oral thesis defense may, in a later semester, make one and only one more attempt to satisfy this requirement.  In doing so, the student may switch between the thesis and the non-thesis option of the program, or between one preliminary exam area and another.

 

Adequate Progress

Normally, a student must be enrolled in at least 5 credits of qualifying courses for each semester (summer excluded) until the degree is awarded.  A student not required to maintain full-time status may register part-time.  Students who wish may enroll in the Time Out Program through the Registrar's Office upon approval of the Faculty Advisor.  Otherwise, the Graduate Committee may remove an inactive student from the degree program.

 

Time Limit

All requirements for the M.S. degree must be completed within four years of the start of graduate studies (see the Catalog for details).

 

 

Ph.D. Degree Requirements

 

General Requirements

Studies leading to the Doctor of Philosophy degree must be chosen so as to contribute to a high level of scholarship in a broad field of study.  Since applied mathematics is by nature interdisciplinary, these will include courses in one or more application areas as well as those within the Department. The Catalog of the Graduate School gives the general requirements for the degree of Doctor of Philosophy.  The following information supplements those requirements.  In all cases not specifically mentioned below, the general requirements as stated in the Catalog are understood to apply.  The program of study must be approved by the Faculty Advisor.

 

Program of Study

A minimum of 60 credits is required for the degree, including 30 in courses numbered 5000 or above and 30 hours of Applied Math dissertation credit. (Neither 4350/5350, 4360/5360, nor 4720/5720 generally count towards this requirement.)  A grade of B- or higher must be attained in each course.  To prepare for the preliminary exams, students are encouraged to take the graduate sequences in applied analysis (5440/5450) and numerics (5600/5610).  An additional graduate sequence in an application area is also recommended.  See the last section for a list of acceptable 5000-level sequences.  Other sequences require Faculty Advisor approval.

 

Students must take at least two semesters of seminar courses (8000, 8100, 8300 or 8600).  These courses are to be taken no earlier than the second year of graduate study in the Department of Applied Mathematics.  First year graduate students are not permitted to enroll in these seminar courses except by special petition to the graduate committee.  Note:  Transcripts might state "repeat—not for credit" when seminar courses are taken more than once.  This statement is an artifact of the system and should be ignored in this case.  Repeated seminars will be credited towards the M.S. or Ph.D.

 

Finally, each student must take a year-long graduate sequence outside of applied mathematics in an area where mathematics has significant application.  Faculty Advisor approval of this sequence is required.

 

Dissertation Hours

Students are required to enroll in a minimum of 30 dissertation hours to complete their Ph.D. requirements.  Students can enroll in a maximum of 10 dissertation hours per semester, and must be enrolled in a minimum of 5 dissertation hours after passing the comprehensive exam.  (Students electing to be part-time after passing the comprehensive exam may enroll in 3+ dissertation hours, but they must be full-time the semester they defend their dissertation.)  No more than 10 hours taken before the comprehensive exam can be counted toward the 30 minimum hours.  Students must be enrolled in 5 dissertation hours the semester they defend their dissertation.

 

Transfer Credit

The Graduate School will allow doctoral students to transfer from another institution up to 21 semester hours of course work toward the Ph.D. degree.  All transfer requests MUST have the approval of the Graduate Committee in Applied Mathematics. Work already applied toward a graduate degree received from CU-Boulder or another institution cannot be accepted for transfer toward another graduate degree of the same level at CU-Boulder.  For example, work already applied to meet requirements for a masterŐs degree earned by a student cannot be used toward a second masterŐs degree from CU.  In addition, work completed for a doctoral degree may not be applied toward a subsequent masterŐs degree.

 

M.S. Degree for Ph.D. Students

Courses taken at the University to satisfy the requirements for the M.S. degree in Applied Mathematics normally will be counted in considering the minimum requirements for the Ph.D. degree in Applied Mathematics.

 

A student pursuing a Ph.D. degree need not receive the M.S. degree.  Any Ph.D. student also wanting to receive the M.S. degree must satisfy the requirements for that degree.  A student may complete the M.S. non-thesis option by passing one of the preliminary exams.  Interested students should contact the Graduate School for the necessary materials.

 

Preliminary Exam

Preliminary exams are offered in four areas: analysis, numerics, partial differential equations, and probability/statistics.  Students must take the numerics and analysis exams, and either one of the other two.  Each is a three-hour written exam.  Guidelines and previous exams are available in the departmental office and on the web to aid in preparation. 

 

The preparatory courses for these prelims are APPM 5440-5450 for analysis, APPM 5600-5610 for numerics, APPM 5470 (not MATH 5470) for partial differential equations, and APPM 5520-5560 for probability/statistics.  Please note that APPM 5520 and APPM 5560 can be taken in any order, and that APPM 5570-5580 is not preparatory for the probability/statistics prelim.

 

Students are expected to complete the preliminary exam requirement successfully prior to the start of their third year, and prior to passing comps and being admitted to candidacy.  Continuation of financial support is contingent upon meeting this requirement.  Exams are given at the beginning of the fall and spring semesters.

 

Each preliminary exam has three possible grades:  Ph.D. Research Pass, Pass, and Fail.  To complete the preliminary exam requirement successfully, a student must pass three exams, including analysis and numerics.  At least two of the passes must be "Ph.D. Research Passes"--the third may be a "Pass".  No exam can be taken more than twice.

 

Foreign Language Requirement

The Department requires that, before advancing to candidacy, all students must provide evidence of competency in a foreign language.  Approved foreign languages include Chinese, French, German, Japanese, Russian, and Spanish, but others may be approved by petitioning the Graduate Committee.  Anyone with a foreign language as a native tongue may be judged by the Graduate Committee to have met this requirement, while others may satisfy the language requirement by having successfully completed a one-year college course.  Alternatives for establishing competency may be proposed to the Graduate Committee.  In any case, providing evidence of competency is the student's responsibility.

 

Thesis Committee

After choosing a field of specialization, the student will present to the Chair of the Graduate Committee for approval a list of five faculty members to serve on her/his Thesis Committee.  All members must have CU graduate faculty status.  At least three of the Committee members must be on the Department faculty; the others can be affiliated faculty members or any other faculty member outside the Department.  The Graduate Committee and the Dean of the Graduate School must approve the make-up of the Thesis Committee.  Changes can be made any time to the Committee membership provided the Graduate Committee and Dean of the Graduate School approve them.  The Chair will serve as the student's thesis advisor and supervise her/his research.  The Chair must be a member of the Department faculty or affiliate faculty, as well as the CU graduate faculty.  If an affiliate serves as Chair, then a member of the Department faculty must serve as an Associate Chair.

 

Comprehensive Exam (Thesis Proposal)

Prior to admission to candidacy, and not later than the end of the third year in the Department, each student must pass the comprehensive exam.  (Students must satisfy the preliminary exam requirement before attempting the comprehensive exam.)  The purposes of the comprehensive exam are to insure that the student has a sufficient grasp of the fundamentals of the chosen thesis area to begin research, the ability to exchange ideas and information with the members of the examining board, and a broad base of knowledge in applied mathematics. 

 

At least two weeks before the exam, an application for admission to candidacy for the Ph.D. must be submitted for approval to the Graduate School. At least one week before the exam, the student must submit a 5-10 page thesis proposal, complete with motivation for the topic and references to key papers, to each member of the Thesis Committee.  This proposal should be written in consultation with the Chair of the Thesis Committee.

 

The exam will consist of a presentation by the student on her/his research proposal of at most one hour in length, followed by a questioning period of up to one additional hour.  The presentation portion is open to all faculty and students in the program.  The student is responsible for ensuring general announcement of the comprehensive exam to the Department at least three days beforehand.  The Thesis Committee will constitute the examining board.  A passing grade is given if at least four of the five members (including the Chair) of the examining committee vote satisfactory performance.

 

 

Thesis Requirement

A thesis must be based on original investigation and reflect a mature understanding and critical judgment of the subject matter, as well as familiarity with tools and methods of research.  The thesis subject must be approved by the Thesis Committee, and must be submitted to the Committee at least 30 days prior to the defense.  The thesis should consist of material that is publishable in a peer-reviewed journal in, or related to, applied mathematics, and it must follow the format prescribed by the Graduate School (see the Catalog).

 

The student is expected to submit a hard copy (bound, on thesis bond) and an electronic copy (in PostScript or PDF format) of the thesis to the Department before graduation.  The electronic version will be placed on the Department web site for public access. The department copy of the dissertation should have title, author and graduation year embossed on the spine.  To help cover this additional expense, the department will reimburse the student up to $30 toward the binding expense.  The Denver Bookbinding Company (http://www.denverbook.com/) provides this service at a reasonable cost.

 

 

The Thesis Defense

After the thesis has been completed, a final exam on the thesis and related topics will be conducted.  This exam is oral and open to the public. The student is responsible for ensuring general announcement of the thesis defense to the Department at least three days beforehand.  The exam is to be administered by the Thesis Committee.  More than one dissenting vote will disqualify the candidate in the final exam.

 

Time Limit

Doctoral students are expected to complete all degree requirements within six years of admission (see the Catalog for details).

 

Grievance Policy

If a student feels that he/she has received unfair treatment academically or as a teaching or research assistant, then the student should first attempt to resolve the problem with the faculty member involved.  If this fails, the student should talk to the Faculty Advisor who will bring the matter before the Graduate Committee.  If the student is dissatisfied with this resolution, he/she may file a written formal complaint with the Associate Dean of the Graduate School.

 

 

CONCURRENT B.S./M.S. DEGREE

 

Purpose of the Program

This is a five-year degree program which leads to both a Bachelor of Science and a Master of Science degree in Applied Mathematics at the end of the five years. It enables well-qualified and motivated students to experience graduate-level course-work earlier in their education and to obtain an M.S. degree in a reduced time period.

 

Consult the Applied Math Undergraduate Curriculum Guide for more information about the B.S./M.S.

 

 

Combined M.S. and M.A. Program with MCD Biology

 

Purpose of the Program

This Program offers two Masters degrees, an M.S. in Applied Mathematics and an M.A. in MCD Biology. This is a three-year interdisciplinary Program. The Department of Applied Mathematics will be the primary program.

 

The goal of the Program is to produce well-trained applied mathematics students who are capable of making serious contributions leading to advancements in molecular biology. Such students will be well educated in computational sciences, statistics, probability, and molecular biology.

 

Admission to the Program

Students are expected to meet all requirements for admission to the graduate program in the Department of Applied Mathematics and possess a basic science background suitable for pursuit of this dual degree. Students are also expected to meet minimum requirements for admission to the graduate program in MCD Biology.  Adequate undergraduate preparation consists of successful completion of basic courses on cell and molecular biology. Any student deemed deficient in either area will be required to take Cell Biology (MCDB 3120) and Molecular Biology (MCDB 3500) after enrollment.

 

Students will be required to apply to both programs, with APPM the primary one.  Subject to joint recommendation and approval by APPM and MCDB, incoming students will be admitted to this Dual Degree Program as a regular part of the Applied Mathematics recruitment process.

 

Interested students should apply to the Applied Mathematics Program and indicate their interest in the concurrent degree on their application.

 

The student will be expected to start the programs simultaneously, except in unusual cases when clearly nothing would be compromised. Continuation of the student beyond the first year is subject to approval by APPM and MCDB.

 

Curriculum

The required curriculum in Applied Mathematics includes the following 7 three-credit courses: Methods of Applied Mathematics: Partial Differential and Integral Equations (APPM 5470), Numerical Analysis I and II (APPM 5600 and 5610), Introduction to Mathematical Statistics (APPM 5520), Statistical Applications Software and Methods (APPM 5580), Numerical Methods for Unconstrained Optimization (CSCI 6676), and two semesters of Independent Study in Applied Mathematics (APPM 6900). One semester of APPM 6900 (2 credits) for this Program will focus on a basic study of the principles of genetics.  The second semester of APPM 6900 (1 credit) will focus on oral student presentations on thesis research, including fielding questions, responding to critiques, and presenting background information. Both sections of APPM 6900 will be arranged in consultation with the student's Faculty Advisor, who will nominally serve as the course instructor.

 

This package of 21 credits provides the necessary background in general applied mathematics, computational mathematics, and statistics/probability for students to address challenging problems at the interface of applied mathematics and biology. This preparation is appropriate for either an academic or commercial setting, especially in the emerging area of bioinformatics.

 

In MCD Biology, the core curriculum in this Program consists of 21 credits as follows. A student takes three 3-credit courses, usually in the second year: Cell Structure and Function (MCDB 5210), Gene Expression (MCDB 5230), and Topics in Cell and Developmental Biology (MCDB 5250). In the third year, a student takes either Molecular Genetics (MCDB 5220), or Cell Signaling and Developmental Regulation (CHEM/MCDB 5426). In either case, the student also takes one 3-credit graduate elective in MCDB and 6 credits of Master's Thesis (MCDB 6950; see below). The graduate elective course can be the other of the two required courses mentioned above.

 

Master's Prelim and Thesis Requirements

Twenty-one credits of core courses are required within each Department. The proposed MCDB courses would fulfill the current APPM requirement of an outside sequence and election of a third course. Similarly, the APPM courses would serve as outside and elective courses to fulfill MCDB requirements.

 

The MasterŐs degree requirements for the APPM M.S. will be fulfilled by the non-thesis (Plan II) option. The requirements for a thesis (Plan I) program will apply to the MCDB masterŐs degree.  Thesis hours count only toward MCDBŐs degree. A student must pass the MCDB Prelim (made up of the exams in MCDB 5210/5230). The student must also successfully complete 6 credits of MCDB 6950 by writing a Master's thesis on original research in an area at the interface between Applied Mathematics and Molecular, Cellular, and Developmental Biology.

 

The Graduate Chairs of both departments must approve successful completion of their respective degree requirements before either degree is confirmed.

 

Leaving the Program

A student will be terminated from the dual degree program if they are terminated from either individual program. The student may petition the APPM Graduate Committee to remain in the APPM program if he/she was terminated from the MCDB program.

 

 

Computational Science and Engineering Track

 

Purpose of the Program

The purpose of this program is to meet the needs of students who want to learn the basic concepts and skills of Computational Science and Engineering, and then to continue towards a Ph.D. in a discipline outside Applied Mathematics.  A student who completes this program successfully will obtain a MasterŐs degree in Applied Mathematics, in the Computational Science and Engineering Track.

 

The program is designed to provide interested students with a foundation in computational mathematics and, at the same time, to allow sufficient latitude for the student to become proficient in an outside discipline. Approximately half of the credits for the MasterŐs degree will be taken from a department other than Applied Mathematics.

 

Admission to the Program

A student in the Computational Science and Engineering Track will be enrolled simultaneously in two graduate programs, one in Applied Mathematics and one in the department from which the student wishes to receive a Ph.D.  An interested student can apply for admission to this Track either when applying for graduate study at CU, or at any time in the studentŐs first two years of graduate study.  First-year and second-year graduate students in any of the participating departments may apply for admission to this program.

 

Curriculum

The proposed curriculum is flexible in that a student may choose from a set of courses those most useful to the discipline in which the Ph.D. is sought.  Different departments will have different sets of appropriate courses.  The specific set of courses for each participating department will be specified in a Memorandum of Understanding between that department and Applied Mathematics.  Four examples of such course listings are given below, for: (i) Physics;  (ii) Astrophysical and Planetary Sciences (APS);  (iii) Aerospace Engineering; and (iv) Atmospheric and Oceanic Sciences (PAOS). 

 

Proposed Curriculum

 

I.   APPM courses - 15 credits:

1. APPM 5600 (3 credits) - Numerical Analysis I

2. APPM 5610 (3 credits) - Numerical Analysis II

3. APPM 5440  (3 credits) - Applied Analysis I

                        (Students who have not had the equivalent of MATH 4310 may need to take this course

before taking APPM 5440.)

            4. One of the following choices, to be approved by the outside advisor:

                        APPM 5470 (3 credits) - Partial Differential Equations

                        APPM 5460 (3 credits) - Dynamical Systems, Differential Equations, and Chaos

                        APPM 5560 (3 credits) - Markov Processes, Queues and Monte Carlo Simulations

                        Other approved APPM courses

5. Thesis Option:

                        APPM 6950 (3 credits) - MS thesis

6. Non-thesis option

                        Graduate electives (6 credits), at least 3 credits in APPM.

(A student taking this option must pass one of the four APPM preliminary exams.)

 

II.  CSCI courses - 3 credits: 

            CSCI 5676 (3 credits) - High Performance Scientific Computing I

            (or other approved CSCI course)

 

III.  Credits in second department  - 12 credits approved by outside advisor

            Thesis option: must include 3 credits of MS thesis in outside department

 

IV.  Students working as TA's in APPM must also take:

            APPM 7400 (1 credit) - Teaching and Learning Seminar

 

V.  At least 30 graduate credits are required by the Graduate School.

 

VI. To waive any of the course requirements:

a)   A student can waive the requirement for APPM 5600 and 5610 by passing the APPM

prelim in Numerical Analysis.

b)   A student can waive the requirement for APPM 5440 by passing the APPM prelim in 

Applied Analysis, or by passing the final exam in APPM 5440 given that year.

c)   A student can waive the requirement for APPM 5470 by passing the APPM prelim in

PDEs.

 

VII. Other recommended courses

            APPM 5430 (3 credits) - Applications of Complex Variables          

            APPM 5480 (3 credits) - Methods of Applied Mathematics: Approximation Methods

            APPM 6610 (3 credits) - Introduction to Numerical Partial Differential Equations

            APPM 7400 (3 credits) - Stochastic Simulations

            CSCI 5686 (3 credits) - High Performance Scientific Computing II

 

 

 

 

SOME SAMPLE PROGRAMS

 

A.  Computational Physics, non-thesis option

Any four of the following six courses, along with the required courses in APPM and CSCI:

PHYS 5250-3.    Quantum Mechanics I

PHYS 5260-3.    Quantum Mechanics II

PHYS 7310-3.    Electromagnetic Theory I

PHYS 7320-3.    Electromagnetic Theory II

PHYS 5210-3.    Classical Mechanics

PHYS 7230-3.    Statistical Mechanics

 

B.    Computational Astrophysics, non-thesis option

Any four of the following eight courses, along with the required courses in APPM and CSCI:

ASTR 5110-3.    Internal Processes in Gases 1

ASTR 5120-3.    Internal Processes in Gases 2

ASTR 5140-3.    Astrophysical and Space Plasmas

ASTR 5150-3.    Introductory Plasma Physics

ASTR 5400-3.    Introduction to Fluid Dynamics

ASTR 5540-3.    Mathematical Methods

ASTR 5560-3.    Radiative Processes in Planetary Atmospheres

ASTR 5820-3.    Origin and Evolution of the Planetary Systems

 

     C.  Computational Aerospace Mechanics, non-thesis option

Fall semester, first year                                                                      

ASEN 5012-3.  Mechanics of Aerospace Structures             

APPM 5470-3.  Partial Differential Equations

APPM 5600-3.  Numerical Analysis I

 

Spring semester, first year

ASEN 5022-3.  Dynamics of Aerospace Structures

APPM 5610-3.  Numerical Analysis II

CSCI 5676-3   High Performance Scientific Computing I

 

Fall semester, second year                                                                  

ASEN 5007-3.  Introduction to Linear Finite Elements

APPM 5440-3.  Applied Analysis I

APPM Elective-3.

 

Spring semester, second year

ASEN Core-3.

ASEN Elective-3

Elective (any department)-3.

 

Note:  A student may replace APPM 5470 (PDEs) with either APPM 5460 (Dynamical Systems) or APPM 5560 (Markov Processes)

 

    
 
     D. Atmospheric and Oceanic Sciences, non-thesis option

Any four of the following eight courses, along with the required courses in APPM and CSCI:

ATOC 5060-3.  Dynamics of the Atmosphere

ATOC 5061-3.  Dynamics of Oceans

ATOC 5220-3.  Nonlinear Dynamics

ATOC 5225-3.  Thermodynamics of Atmospheres and Oceans

ATOC 5400-3.  Introduction to Fluid Dynamics

ATOC 5410-3.  Fluid Instabilities, Waves, and Turbulence

ATOC 5560-3.  Radiative Processes in Planetary Atmospheres

ATOC 6100-3.  Predicting Weather and Climate

 

Master's Requirements

In accordance with the rules of CUŐs Graduate School, credits from a given graduate course can be applied only towards one MasterŐs degree.  Specifically, if graduate credits from a participating department are used to obtain a MasterŐs degree in Applied Mathematics in the Computational Science and Engineering Track, then those same credits cannot also be used to obtain a MasterŐs degree from the other department.  The credits can be applied towards a Ph. D. from either department.

 

The program is self-contained.  If desired, a student can finish with a MasterŐs degree in Applied Mathematics, and embark on a career at the MasterŐs level.

 

 

APPM Teacher Licensure Option

 

Every graduate student in the Department of Applied Mathematics takes a year-long sequence of courses in some area of application of mathematics.  One option is to take this sequence in the School of Education, and ultimately to obtain both a MasterŐs degree in Applied Mathematics and pursue a license to teach mathematics in a secondary school (i.e., in middle through high school).  This option is not simple, and pursuing it will delay graduation from the department.  Nevertheless, for graduate students in Applied Mathematics who also seek a teaching license, here are some guidelines.

 

The Teacher Education Program (TEP) in the School of Education for Secondary Mathematics Teacher Licensure consists of seven courses (EDUC 3013-4, EDUC 3023-4, EDUC 4112-3, EDUC 4122-3, EDUC 4232-3, EDUC 6804-3, EDUC 5375-4), plus one semester of student teaching (which includes EDUC 4512-2 and 4712-12, and is a full-time full-semester in-school commitment), plus a passing score on the PRAXIS II or PLACE licensure exam in mathematics.  However, only students who have been admitted to TEP are eligible to enroll in all but the first of these courses.  Before being admitted to TEP, a student must have met requirements that include a minimum of 56 hours of college work, grade point average minimums (2.75 in several areas, including mathematics courses), 25 hours of youth experience, and a passing score on the PRAXIS II or PLACE licensure exam in mathematics.  Prior to student teaching, the student must have completed all but the student teaching semester courses, a series of mathematics requirements, and the PRAXIS II or PLACE Basic Skill Assessment.  How much preliminary work will be required depends on details of the student's previous college background.  Interested students should consult with an advisor in the School of Education in order to learn precise details.  A student interested in pursuing this dual-objective program should plan to complete all of the necessary courses within APPM in the first two years.  Some of the courses required by the School of Education may also be taken during this time.  With very careful planning from the beginning of the APPM program, it is potentially possible to complete both the APPM and TEP within three to three and one-half years.  A student following this plan will not necessarily receive financial assistance (in the form of a TA) from APPM after the second year.

 

Interested students should consult with an advisor in the School of Education in order to learn precise details. Dana Runge (Education, room 151, 303-492-2559, Dana.Runge@Colorado.EDU) is the most appropriate contact in the School of Education.  The APPM requirement of a year-long sequence outside the Department can be satisfied by taking EDUC 4372, plus either EDUC 4232 or EDUC 4122.  The other requirements for a Master of Science in Applied Mathematics are found in the previous sections of this supplement.

 

 

Graduate Courses

 

Numerous courses in other departments at the University, as well as the University of Colorado at Denver, are, in essence, courses in applied mathematics and can be taken for graduate credit in Applied Mathematics.  In fact, each graduate student must take a year-long sequence outside the department.  Consult the Faculty Advisor for more information and for approval.

 

 

 

Basic Courses

 

Acceptable 5000-level APPM sequences include the following (others require Faculty Advisor approval):

5430-5470, 5440-5450, 5460-5470, 5470-5480, 5520-5540, 5520-5560, 5570-5580, and 5600-5610.

 

The following courses that are crosslisted as graduate/undergraduate courses do not count towards the 30 credit hour M.S. or Ph.D. requirement:

APPM 5350 (3) Methods in Applied Mathematics: Fourier Series and Boundary Value Problems

APPM 5360 (3) Methods In Applied Mathematics: Complex Variables and Applications

APPM 5720 (3) Open Topics In Applied Mathematics

 

All of the remaining courses listed below do count towards the 30 credit hour MS or Ph.D. requirement.

 

APPM 5120 (3).  Introduction to Operations Research.  Studies linear and nonlinear programming, the simplex method, duality, sensitivity, transportation and network flow problems, some constrained and unconstrained optimization theory, and the Kuhn-Tucker conditions as time permits.  Prereqs.:  APPM 3310 or MATH 3130.  Same as APPM 4120 and MATH 4120/5120.  (Normally offered spring semester.)

 

APPM 5380 (3).  Modeling in Applied Mathematics.  An exposition of a variety of mathematical models arising in the physical and biological sciences.  Students carry out modeling projects and present these in class.  Topics can vary:  GPS navigation, medical imaging, ocean waves, computerized facial recognition.  Prereqs.: APPM 2350 and 2360.  Recommended: APPM 3310, 4350, and 4650.  Same as APPM 4380.  (Normally offered fall semester.)

 

APPM 5430 (3).  Methods in Applied Mathematics: Applications of Complex Variables.  Reviews basic ideas of complex analysis, including solutions of ODE's and PDE's of physical interest via complex analysis; conformal mapping including Schwarz-Christoffel transformations and generalizations; computational methods; Riemann-Hilbert problems; and topics in asymptotic methods.  Prereq.: APPM 4360 or 5360, or instructor consent.  (Offered on a variable schedule.)

 

APPM 5440 (3).  Applied Analysis 1.  Discusses the elements of basic real and complex analysis, Banach spaces, LP spaces, and many relevant inequalities.  Includes applications of existence and uniqueness of solutions to various types of ordinary differential equations, partial differential equations and integral equations.  Prereqs.: MATH 4310 and 4320 or equivalent; MATH 3130 or equivalent; or instructor consent.  (Normally offered fall semester.)

 

APPM 5450 (3).  Applied Analysis 2.  Continuation of APPM 5440. Prereqs.: APPM 5440 or instructor consent.  (Normally offered spring semester.)

 

APPM 5460 (3).  Methods in Applied Mathematics: Dynamical Systems and Differential Equations and Chaos.  Introduces the theory and applications of dynamical systems through solutions of differential equations. Covers existence and uniqueness theory, local stability properties, qualitative analysis, global phase portraits, perturbation theory, and bifurcation theory. Special topics may include Melnikov methods, averaging methods, bifurcations to chaos, and Hamiltonian systems.  Prereqs.: undergraduate courses equivalent to APPM 2360, 3310, and MATH 4310.  (Normally offered spring semester of even years.)

 

APPM 5470 (3).  Methods of Applied Mathematics: Partial Differential and Integral Equations.  Studies properties and solutions of partial differential equations.  Covers methods of characteristics for partial differential equations, well-posed problems, main results for the well-known equations, GreenŐs functions, and related integral equations.  Prereqs.: APPM 4350 and 4360 or MATH 4430 or equivalent.  Same as MATH 5470.  (Normally offered fall semester.)

 

APPM 5480 (3).  Methods of Applied Mathematics: Approximation Methods.  Covers asymptotic evaluation of integrals (stationary phase and steepest descent), perturbation methods (regular and singular methods, and inner and outer expansions), multiple scale methods, and applications to differential and integral equations.  Prereqs.: APPM 5470 or instructor consent.  (Normally offered spring semester of odd years.)

 

APPM 5520 (3).  Introduction to Mathematical Statistics.  Examines point and confidence interval estimation.  Principles of maximum likelihood sufficiency and completeness; tests of simple and composite hypotheses, linear models, and multiple regression analysis.  Analyzes variance distribution-free methods.  Prereqs.: one semester calculus–based probability such as MATH 4510 or APPM 3570.  Same as APPM 4520 and MATH 4520/5520.  (Normally offered spring semester.)

 

APPM 5540 (3).  Introduction to Time Series.  Single and multivariable regression, forecasting using regression models, time series models, and modeling with MA, AR, ARMA, and ARIMA models, forecasting with time series models, and spectral analysis.  Prereqs.:  APPM 3570 or MATH 4510, and APPM 5520/MATH 5520.  Same as APPM 4540 and MATH 4540/5540.  (Normally offered spring semester.)

 

APPM 5560 (3).  Markov Processes, Queues and Monte Carlo Simulations.  Brief review of conditional probability and expectation followed by a study of Markov chains, both discrete and continuous time.  Queuing theory, terminology, and single queue systems are studied with some introduction to networks of queues.  Uses Monte Carlo simulation of random variables throughout the semester to gain insight into the processes under study.  Prereqs.: APPM 3570 or equivalent.  Same as APPM 4560.  (Normally offered fall semester.)

 

APPM 5570 (3).  Statistical Methods.  Covers discrete and continuous probability laws, random variables; expectations; laws of large numbers and central limit theorem; estimation, testing hypothesis, analysis of variance, regression analysis, nonparametric methods.  Emphasizes applications with an introduction to packaged computer programs.  Prereqs.: APPM 1360 or equivalent CALC 2 course.  Same as APPM 4570.  (Normally offered fall and spring semester.)

 

APPM 5580 (3).  Statistical Applications:  Software and Methods.  Continuation of APPM 5570.  Combines statistical methods with practical applications and computer software.  Develops commonly used statistical models such as analysis of variance as well as linear and logistic regression.  The statistical models are implemented and interpreted in the context of actual data sets using available statistical software.  Prereqs.: one semester of statistics.  Same as APPM 4580.  (Normally offered spring semester.)

 

APPM 5600 (3).  Numerical Analysis 1.  Solution of nonlinear algebraic equations, interpolation, approximation theory, and numerical integration.  Prereqs.: APPM 3130 or MATH 3130, and experience with a scientific programming language.  Same as MATH 5600.  (Normally offered fall semester.)

 

APPM 5610 (3).  Numerical Analysis 2.  Solution of linear systems, eigenvalue problems, optimization problems, and ordinary and partial differential equations.  Prereqs.: APPM 5600 or MATH 5600.  Same as MATH 5610.  (Normally offered spring semester.)

 

Advanced Courses

 

APPM 6520 (3).  Mathematical Statistics.  Emphasizes mathematical theory of statistics.  Topics include distribution theory, estimation and testing of hypotheses, multivariate analysis, and nonparametric inference, all with emphasis on theory.  Prereqs.: APPM 5520 or MATH 5520.  Same as MATH 6520.  (Offered on a variable schedule.)

 

APPM 6550 (3).  Introduction to Stochastic Processes.  Systematic study of Markov chains and some of the simpler Markov processes, including renewal theory, limit theorems for Markov chains, branching processes, queuing theory, birth and death processes, and Brownian motion. Applications to physical and biological sciences.  Prereqs.: MATH 4310, MATH 4510 or APPM 3570, or APPM 4560, or instructor consent.  Same as MATH 6550.  (Normally offered by MATH in spring semester of even years.)

 

APPM 6610 (3).  Introduction to Numerical Partial Differential Equations.  Covers finite difference, finite element, finite volume, pseudo-spectral, and spectral methods for elliptic, parabolic, and hyperbolic partial differential equations.  Prereq.: APPM 5600.  Recommended prereq.: APPM 5610 or graduate numerical linear algebra. (Normally offered fall semesters of odd years.)

 

APPM 6620 (3).  Numerical Computation in Applied Mathematics 1.  Advanced topics in the numerical solution of ordinary and partial differential equations, initial and boundary value problems, and stability and convergence of difference schemes.  Prereqs.: APPM 4650/MATH 4650, or instructor consent.  (Offered on a variable schedule.)

 

APPM 6630 (3).  Numerical Computation in Applied Mathematics 2.  Continuation of APPM 6620.  Prereqs.: APPM 6620 or instructor consent.  (Offered on a variable schedule.)

 

APPM 6900 (1-3).  Independent Study.  Introduces graduate students to research focuses of the Department of Applied Mathematics.  Prereq.: instructor consent.

 

APPM 6940 (1-3).  MasterŐs Degree Candidate. 

 

APPM 6950 (1-6).  MasterŐs Thesis.  May be repeated up uo 12 total credit hours.

 

APPM 7100 (3).  Mathematical Methods in Dynamical Systems.  Covers dynamical systems defined by mappings and differential equations.  Hamiltonian mechanics, action-angle variables, results from KAM and bifurcation theory, phase plane analysis, Melnikov theory, strange attractors, chaos, etc.  Prereq.: APPM 5460.  (Offered on a variable schedule.)

 

APPM 7300 (3).  Nonlinear Waves and Integrable Equations.  Includes basic results associated with linear dispersive wave systems, first-order nonlinear wave equations, nonlinear dispersive wave equations, solitons, and the method of the inverse scattering transform.  Prereqs.: APPM 5470-5480, PHYS 5210, or instructor consent.  (Offered on a variable schedule.)

 

APPM 7400 (3).  Topics in Applied Mathematics.  Provides a vehicle for the development and presentation of new topics with the potential of being incorporated into the core courses in applied mathematics.  May be repeated for a total of 6 credit hours.  Prereqs.: instructor consent.

 

APPM 7900 (1-3).  Reading and Research in Applied Mathematics.  Introduces graduate students to research focuses of the Department of Applied Mathematics.  Prereqs.: instructor consent.

 

APPM 8000 (1).  Colloquium in Applied Mathematics.  Introduces graduate students to the research focuses of the Department of Applied Mathematics.  Prereqs.: instructor consent. (Normally offered fall and spring semesters.)

 

APPM 8100 (1).  Seminar in Dynamical Systems.  Introduces advanced topics and research in dynamical systems.  Prereqs.: Instructor consent. (Normally offered fall and spring semesters.)

 

APPM 8300  (1).  Seminar in Applied Mathematics.  Introduces advanced topics and current research in the area of applied mathematics.  The seminar will also provide the opportunity for students to read articles prior to seminar lectures  Prereqs.:  Graduate standing in Applied Mathematics.

 

APPM 8600 (1).  Seminar in Computational Mathematics.  Introduces advanced topics and research in computational mathematics.  Prereqs.: Instructor consent.  (Normally offered fall and spring semesters.)

 

APPM 8990 (1-10).  Doctoral Dissertation.  All doctoral students must register for no fewer than 30 hours of dissertation credit as part of the requirements for the degree.  No more than 10 credit hours may be taken in any one semester.

 

Note:  Transcripts might state "repeat - not for credit" when seminar courses are taken more than once.  This statement is an artifact of the system and should be ignored in this case.  Repeated seminars will be credited towards the MS or Ph.D.